lib/divdf3.c

//===-- lib/divdf3.c - Double-precision division ------------------*- C -*-===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// This file implements double-precision soft-float division
// with the IEEE-754 default rounding (to nearest, ties to even).
//
// For simplicity, this implementation currently flushes denormals to zero.
// It should be a fairly straightforward exercise to implement gradual
// underflow with correct rounding.
//
//===----------------------------------------------------------------------===//

#define DOUBLE_PRECISION
#include "fp_lib.h"

fp_t __divdf3(fp_t a, fp_t b) {
    
    const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
    const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
    const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
    
    rep_t aSignificand = toRep(a) & significandMask;
    rep_t bSignificand = toRep(b) & significandMask;
    int scale = 0;
    
    // Detect if a or b is zero, denormal, infinity, or NaN.
    if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) {
        
        const rep_t aAbs = toRep(a) & absMask;
        const rep_t bAbs = toRep(b) & absMask;
        
        // NaN / anything = qNaN
        if (aAbs > infRep) return fromRep(toRep(a) | quietBit);
        // anything / NaN = qNaN
        if (bAbs > infRep) return fromRep(toRep(b) | quietBit);
        
        if (aAbs == infRep) {
            // infinity / infinity = NaN
            if (bAbs == infRep) return fromRep(qnanRep);
            // infinity / anything else = +/- infinity
            else return fromRep(aAbs | quotientSign);
        }
        
        // anything else / infinity = +/- 0
        if (bAbs == infRep) return fromRep(quotientSign);
        
        if (!aAbs) {
            // zero / zero = NaN
            if (!bAbs) return fromRep(qnanRep);
            // zero / anything else = +/- zero
            else return fromRep(quotientSign);
        }
        // anything else / zero = +/- infinity
        if (!bAbs) return fromRep(infRep | quotientSign);
        
        // one or both of a or b is denormal, the other (if applicable) is a
        // normal number.  Renormalize one or both of a and b, and set scale to
        // include the necessary exponent adjustment.
        if (aAbs < implicitBit) scale += normalize(&aSignificand);
        if (bAbs < implicitBit) scale -= normalize(&bSignificand);
    }
    
    // Or in the implicit significand bit.  (If we fell through from the
    // denormal path it was already set by normalize( ), but setting it twice
    // won't hurt anything.)
    aSignificand |= implicitBit;
    bSignificand |= implicitBit;
    int quotientExponent = aExponent - bExponent + scale;
    
    // Align the significand of b as a Q31 fixed-point number in the range
    // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
    // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2.  This
    // is accurate to about 3.5 binary digits.
    const uint32_t q31b = bSignificand >> 21;
    uint32_t recip32 = UINT32_C(0x7504f333) - q31b;
    
    // Now refine the reciprocal estimate using a Newton-Raphson iteration:
    //
    //     x1 = x0 * (2 - x0 * b)
    //
    // This doubles the number of correct binary digits in the approximation
    // with each iteration, so after three iterations, we have about 28 binary
    // digits of accuracy.
    uint32_t correction32;
    correction32 = -((uint64_t)recip32 * q31b >> 32);
    recip32 = (uint64_t)recip32 * correction32 >> 31;
    correction32 = -((uint64_t)recip32 * q31b >> 32);
    recip32 = (uint64_t)recip32 * correction32 >> 31;
    correction32 = -((uint64_t)recip32 * q31b >> 32);
    recip32 = (uint64_t)recip32 * correction32 >> 31;
    
    // recip32 might have overflowed to exactly zero in the preceeding
    // computation if the high word of b is exactly 1.0.  This would sabotage
    // the full-width final stage of the computation that follows, so we adjust
    // recip32 downward by one bit.
    recip32--;
    
    // We need to perform one more iteration to get us to 56 binary digits;
    // The last iteration needs to happen with extra precision.
    const uint32_t q63blo = bSignificand << 11;
    uint64_t correction, reciprocal;
    correction = -((uint64_t)recip32*q31b + ((uint64_t)recip32*q63blo >> 32));
    uint32_t cHi = correction >> 32;
    uint32_t cLo = correction;
    reciprocal = (uint64_t)recip32*cHi + ((uint64_t)recip32*cLo >> 32);
    
    // We already adjusted the 32-bit estimate, now we need to adjust the final
    // 64-bit reciprocal estimate downward to ensure that it is strictly smaller
    // than the infinitely precise exact reciprocal.  Because the computation
    // of the Newton-Raphson step is truncating at every step, this adjustment
    // is small; most of the work is already done.
    reciprocal -= 2;
    
    // The numerical reciprocal is accurate to within 2^-56, lies in the
    // interval [0.5, 1.0), and is strictly smaller than the true reciprocal
    // of b.  Multiplying a by this reciprocal thus gives a numerical q = a/b
    // in Q53 with the following properties:
    //
    //    1. q < a/b
    //    2. q is in the interval [0.5, 2.0)
    //    3. the error in q is bounded away from 2^-53 (actually, we have a
    //       couple of bits to spare, but this is all we need).
    
    // We need a 64 x 64 multiply high to compute q, which isn't a basic
    // operation in C, so we need to be a little bit fussy.
    rep_t quotient, quotientLo;
    wideMultiply(aSignificand << 2, reciprocal, &quotient, &quotientLo);
    
    // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
    // In either case, we are going to compute a residual of the form
    //
    //     r = a - q*b
    //
    // We know from the construction of q that r satisfies:
    //
    //     0 <= r < ulp(q)*b
    // 
    // if r is greater than 1/2 ulp(q)*b, then q rounds up.  Otherwise, we
    // already have the correct result.  The exact halfway case cannot occur.
    // We also take this time to right shift quotient if it falls in the [1,2)
    // range and adjust the exponent accordingly.
    rep_t residual;
    if (quotient < (implicitBit << 1)) {
        residual = (aSignificand << 53) - quotient * bSignificand;
        quotientExponent--;
    } else {
        quotient >>= 1;
        residual = (aSignificand << 52) - quotient * bSignificand;
    }
    
    const int writtenExponent = quotientExponent + exponentBias;
    
    if (writtenExponent >= maxExponent) {
        // If we have overflowed the exponent, return infinity.
        return fromRep(infRep | quotientSign);
    }
    
    else if (writtenExponent < 1) {
        // Flush denormals to zero.  In the future, it would be nice to add
        // code to round them correctly.
        return fromRep(quotientSign);
    }
    
    else {
        const bool round = (residual << 1) > bSignificand;
        // Clear the implicit bit
        rep_t absResult = quotient & significandMask;
        // Insert the exponent
        absResult |= (rep_t)writtenExponent << significandBits;
        // Round
        absResult += round;
        // Insert the sign and return
        const double result = fromRep(absResult | quotientSign);
        return result;
    }
}